Question 1

Evaluate .

Accepted Answer

A)    
B)    
C)    
D)    
E)  series diverges 
A)    
B)    
C)    
D)    
E)  series diverges

Question 2

Find the Fourier cosine series of f(t) = 2t -   on [0 , 1].

Accepted Answer

A)    -     
B)    -     
C)    -     
D)    -     
E)    - 4   
A)    -     
B)    -     
C)    -     
D)    -     
E)    - 4

Question 3

Find the limit of the sequence   .

Accepted Answer

A)  0 
B)  1 
C)  2 
D)  3 
E)  -1 
A)  0 
B)  1 
C)  2 
D)  3 
E)  -1

Question 4

For what range of x can cos x be replaced by the first three nonzero terms of its Maclaurin series to ensure the error is less than 0.00005?

Accepted Answer

A)    $\approx$ 0.144 
B)    $\approx$ 0.072 
C)    $\approx$ 0.036 
D)    $\approx$ 0.288 
E)   $\approx$ 0.575 
A)    $\approx$ 0.144 
B)    $\approx$ 0.072 
C)    $\approx$ 0.036 
D)    $\approx$ 0.288 
E)   $\approx$ 0.575

Question 5

Calculate the Maclaurin series for the function F(x) =   dt and calculate F(1) correct to three decimal places.

Accepted Answer

A)  F(x) =     , F(1) $\approx$ 0.991 
B)  F(x) =     , F(1) $\approx$ 0.844 
C)  F(x) =     , F(1) $\approx$ 0.863 
D)  F(x) =     , F(1) $\approx$ 0.842 
E)  F(x) =     , F(1) $\approx$ 0.873 
A)  F(x) =     , F(1) $\approx$ 0.991 
B)  F(x) =     , F(1) $\approx$ 0.844 
C)  F(x) =     , F(1) $\approx$ 0.863 
D)  F(x) =     , F(1) $\approx$ 0.842 
E)  F(x) =     , F(1) $\approx$ 0.873

Question 6

Accepted Answer

A)    
B)    
C)  $\infty$ 
D)    
E)    
A)    
B)    
C)  $\infty$ 
D)    
E)

Question 7

Accepted Answer

A) True 
 B)False

Question 8

Evaluate   .

Accepted Answer

A)  -1 
B)  0 
C)  1 
D)    
E)  no sum, series diverges 
A)  -1 
B)  0 
C)  1 
D)    
E)  no sum, series diverges

Question 9

Find the Fourier series of the function f having period 2$\pi$  and defined by f(x) = 0 if -$\pi$ < x $\le$ 0 and f(x) = 1 if 0 < x   $\pi$.

Accepted Answer

A)    +   sin((2n - 1)x) 
B)  1 +   sin((2n - 1)x) 
C)    +   sin(2nx) 
D)  1 +   sin(2nx) 
E)    +   cos((2n - 1)x) 
A)    +   sin((2n - 1)x) 
B)  1 +   sin((2n - 1)x) 
C)    +   sin(2nx) 
D)  1 +   sin(2nx) 
E)    +   cos((2n - 1)x)

Question 10

Which of the following descriptors apply to the sequence   ?
(a) increasing (or ultimately increasing)
(b) decreasing (or ultimately decreasing)
(c) positive (or ultimately positive)
(d) negative (or ultimately negative)
(e) bounded below only
(f) bounded above only
(g) bounded 
(h) unbounded above and below
(i) alternating
(j) divergent (but not to $\infty$ or -$\infty$)
(k) divergent to $\infty$
(l) divergent to -$\infty$
(m) convergent

Accepted Answer

A)  (a), (c), (e), (m) 
B)  (a), (c), (f), (j) 
C)  (a), (c), (g), (m) 
D)  (b), (c), (g), (l) 
E)  (a), (c), (e), (k) 
A)  (a), (c), (e), (m) 
B)  (a), (c), (f), (j) 
C)  (a), (c), (g), (m) 
D)  (b), (c), (g), (l) 
E)  (a), (c), (e), (k)

Question 11

For what values of x does the series  
 (a) converge absolutely? 
(b) converge conditionally?

Accepted Answer

A)  (a) x = 0 only, $~~~~~~~~$(b) nowhere 
B)  (a) -1 < x < 1, $~~~~~~~~$(b) nowhere 
C)  (a) -1 < x < 1, $~~~~~~~~$(b) x = -1 and x = 1 
D)  (a) all real x, $~~~~~~~~$(b) nowhere 
E)  (a) -1 < x < 1, $~~~~~~~~$(b) x = 1 
A)  (a) x = 0 only, $~~~~~~~~$(b) nowhere 
B)  (a) -1 < x < 1, $~~~~~~~~$(b) nowhere 
C)  (a) -1 < x < 1, $~~~~~~~~$(b) x = -1 and x = 1 
D)  (a) all real x, $~~~~~~~~$(b) nowhere 
E)  (a) -1 < x < 1, $~~~~~~~~$(b) x = 1

Question 12

If   diverges to infinity and   $\neq$ 0 for all n, then   converges.

Accepted Answer

A) True 
 B)False

Question 13

Let   =   , n = 1, 2, 3,...
Which of the following statements about the sequence   is true?

Accepted Answer

A)  The sequence is decreasing with a lower bound equal to .   
B)  The sequence is increasing with an upper bound equal to .   
C)  The sequence is increasing with a lower bound equal to 0. 
D)  The sequence is not monotonic and also unbounded. 
E)  The sequence is decreasing with an upper bound equal to .   
A)  The sequence is decreasing with a lower bound equal to .   
B)  The sequence is increasing with an upper bound equal to .   
C)  The sequence is increasing with a lower bound equal to 0. 
D)  The sequence is not monotonic and also unbounded. 
E)  The sequence is decreasing with an upper bound equal to .

Question 14

For what values of x does the series    
 converge?

Accepted Answer

A)  on the interval [1, $\infty$) 
B)  on the intervals (-$\infty$, -1] and [1, $\infty$) 
C)  for all x $\neq$0 
D)  for x = 2 only 
E)  on the interval (-$\infty$, -1] 
A)  on the interval [1, $\infty$) 
B)  on the intervals (-$\infty$, -1] and [1, $\infty$) 
C)  for all x $\neq$0 
D)  for x = 2 only 
E)  on the interval (-$\infty$, -1]

Question 15

The series   converges.

Accepted Answer

A) True 
 B)False

Question 16

For what values of x does the series  
 (a) converge absolutely? 
(b) converge conditionally?

Accepted Answer

A)  (a) on the interval (-2, -1), $~~~~~~~~$(b) at x = -2 and x = -1 
B)  (a) on the interval [-2, -1] $~~~~~~~~$(b) nowhere 
C)  (a) on the interval (-2, -1), $~~~~~~~~$(b) for x < -2 
D)  (a) on the interval (-3, 0), $~~~~~~~~$(b) at x = -3 and x = 0 
E)  (a) on the interval [-2, -1), $~~~~~~~~$(b) at x = -1 
A)  (a) on the interval (-2, -1), $~~~~~~~~$(b) at x = -2 and x = -1 
B)  (a) on the interval [-2, -1] $~~~~~~~~$(b) nowhere 
C)  (a) on the interval (-2, -1), $~~~~~~~~$(b) for x < -2 
D)  (a) on the interval (-3, 0), $~~~~~~~~$(b) at x = -3 and x = 0 
E)  (a) on the interval [-2, -1), $~~~~~~~~$(b) at x = -1

Question 17

The P-series   , x > 0 converges for all x such that:

Accepted Answer

A) x > 1 B) x $\ge$ 1 C) 0 < x < 1 D) 0 < x $\le$ 1 E) x (- $\infty$ , $\infty$) A) x > 1 B) x $\ge$ 1 C) 0 < x < 1 D) 0 < x $\le$ 1 E) x (- $\infty$ , $\infty$)

Question 18

The series   is an alternating series.

Accepted Answer

A) True 
 B)False

Question 19

If   is conditionally convergent, then   must be divergent.

Accepted Answer

A) True 
 B)False

Question 20

Use appropriate Maclaurin series to evaluate .

Accepted Answer

A)  -1 
B)  $\infty$ 
C)  3 
D)  1 
E)  -$\infty$ 
A)  -1 
B)  $\infty$ 
C)  3 
D)  1 
E)  -$\infty$

Evaluate .

Find the Fourier cosine series of f(t) = 2t - on [0 , 1].

Find the limit of the sequence .

For what range of x can cos x be replaced by the first three nonzero terms of its Maclaurin series to ensure the error is less than 0.00005?

Calculate the Maclaurin series for the function F(x) = dt and calculate F(1) correct to three decimal places.

Evaluate .

Find the Fourier series of the function f having period 2 $\pi$ and defined by f(x) = 0 if - $\pi$ < x $\le$ 0 and f(x) = 1 if 0 < x $\pi$ .

For what values of x does the series (a) converge absolutely? (b) converge conditionally?

If $If diverges to infinity and \neq 0 for all n, then converges.$ diverges to infinity and $If diverges to infinity and \neq 0 for all n, then converges.$ $\neq$ 0 for all n, then $If diverges to infinity and \neq 0 for all n, then converges.$ converges.

Let = , n = 1, 2, 3,... Which of the following statements about the sequence is true?

For what values of x does the series converge?

The series converges.

For what values of x does the series (a) converge absolutely? (b) converge conditionally?

The P-series , x > 0 converges for all x such that:

The series is an alternating series.

If is conditionally convergent, then must be divergent.

Use appropriate Maclaurin series to evaluate .

Preliminaries

Limits and Continuity

Differentiation

Transcendental Functions

More Applications of Differentiation

Integration

Techniques of Integration

Applications of Integration

Conics, Parametric Curves, and Polar Curves

Vectors and Coordinate Geometry in 3-Space

Vector Functions and Curves

Partial Differentiation

Applications of Partial Derivatives

Multiple Integration

Vector Fields

Vector Calculus

Differential Forms and Exterior Calculus

Ordinary Differential Equations

Filters

Exam 10: Sequences, Series, and Power Series

Evaluate .

Find the Fourier cosine series of f(t) = 2t - on [0 , 1].

Find the limit of the sequence .

For what range of x can cos x be replaced by the first three nonzero terms of its Maclaurin series to ensure the error is less than 0.00005?

Calculate the Maclaurin series for the function F(x) = dt and calculate F(1) correct to three decimal places.

Evaluate .

Find the Fourier series of the function f having period 2 π\piπ and defined by f(x) = 0 if - π\piπ < x ≤\le≤ 0 and f(x) = 1 if 0 < x π\piπ .

For what values of x does the series (a) converge absolutely? (b) converge conditionally?

If diverges to infinity and ≠\neq= 0 for all n, then converges.

Let = , n = 1, 2, 3,... Which of the following statements about the sequence is true?

For what values of x does the series converge?

The series converges.

For what values of x does the series (a) converge absolutely? (b) converge conditionally?

The P-series , x > 0 converges for all x such that:

The series is an alternating series.

If is conditionally convergent, then must be divergent.

Use appropriate Maclaurin series to evaluate .

Preliminaries

Limits and Continuity

Differentiation

Transcendental Functions

More Applications of Differentiation

Integration

Techniques of Integration

Applications of Integration

Conics, Parametric Curves, and Polar Curves

Vectors and Coordinate Geometry in 3-Space

Vector Functions and Curves

Partial Differentiation

Applications of Partial Derivatives

Multiple Integration

Vector Fields

Vector Calculus

Differential Forms and Exterior Calculus

Ordinary Differential Equations

Filters

Find the Fourier series of the function f having period 2 $\pi$ and defined by f(x) = 0 if - $\pi$ < x $\le$ 0 and f(x) = 1 if 0 < x $\pi$ .

If $If diverges to infinity and \neq 0 for all n, then converges.$ diverges to infinity and $If diverges to infinity and \neq 0 for all n, then converges.$ $\neq$ 0 for all n, then $If diverges to infinity and \neq 0 for all n, then converges.$ converges.